How to find range of an absolute value equation?

Understanding how to find the range of an absolute value equation is essential in solving various mathematical problems. The range represents the set of possible output values that the equation can produce. In other words, it defines the vertical extent of the graphed equation. By following a few simple steps, you can easily determine the range of an absolute value equation.

Solving an Absolute Value Equation

To find the range of an absolute value equation, follow these steps:

Step 1: Recognize the Absolute Value Function

The absolute value function is denoted by || and is commonly represented by f(x) = |x|. It takes any real number x and makes it positive or zero. Understanding this basic concept is vital in finding the range.

Step 2: Consider the Given Equation

To find the range of a specific absolute value equation, consider the given equation carefully. For instance, let’s consider the equation f(x) = |2x – 3| – 1.

Step 3: Isolate the Absolute Value

To proceed, isolate the absolute value part of the equation. In this case, the absolute value is |2x – 3|.

Step 4: Set Up Two Equations

Since the absolute value equation can result in two possible outputs, we must set up two separate equations to cover each case, one where the expression inside the absolute value is positive and one where it is negative.

Step 5: Solve for x in Each Equation

Solve each equation for x separately. For instance, in the case where 2x – 3 is positive, we have 2x – 3 = 0, which simplifies to x = 3/2. When 2x – 3 is negative, we have 2x – 3 = -0, leading to 2x – 3 = 0, and x = 3/2 as well.

Step 6: Determine the Range

Now that we have the values of x, we need to find the corresponding y-values to identify the range. Plug in the x-values obtained in previous steps back into the original equation, f(x) = |2x – 3| – 1.

By substituting x = 3/2 in the equation, we find f(3/2) = |2(3/2) – 3| – 1, which simplifies to f(3/2) = |3 – 3| – 1 = |-1| – 1 = 1 – 1 = 0.

Therefore, the range of the equation f(x) = |2x – 3| – 1 is {0}.

Related FAQs

1. What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value function, denoted by ||.

2. How does the absolute value function work?

The absolute value function takes a real number x and returns a positive or zero value, regardless of its original sign.

3. Can an absolute value be negative?

No, the absolute value of a number cannot be negative. It always returns a non-negative value.

4. Why do we isolate the absolute value in the equation?

Isolating the absolute value allows us to analyze two cases – one where the expression inside the absolute value is positive and another where it is negative.

5. Are there always two possible outputs for an absolute value equation?

Yes, the absolute value equation can yield two possible outputs, depending on the sign of the expression inside the absolute value.

6. Can the range of an absolute value equation be an empty set?

Yes, if both cases in the equation result in an absolute value of zero, the range will be an empty set.

7. Can the range of an absolute value equation contain multiple values?

No, the range of an absolute value equation usually contains a single value or an empty set.

8. Is it possible for the range of an absolute value equation to be all real numbers?

Yes, if one of the cases in the equation results in an absolute value of zero, the range will be all real numbers.

9. What happens if the coefficient of x is negative in an absolute value equation?

If the coefficient of x is negative, we need to reverse the inequality when solving for x.

10. Is it necessary to find the range of an absolute value equation?

Determining the range of an absolute value equation is crucial when solving related problems or graphing equations.

11. Can we find the range of an absolute value equation without isolating the absolute value?

No, isolating the absolute value is essential in finding the range of an absolute value equation.

12. Are there any restrictions on the type of absolute value equations for which we can find the range?

No, we can find the range of any absolute value equation as long as we correctly follow the necessary steps.

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